Evaluating the performance of the skewed distributions  

to forecast Value at Risk in the Global Financial Crisis* 

 

Pilar Abad 
Universidad Rey Juan Carlos and IREA-RFA 
Paseo Artilleros s/n.  
28032, Madrid (Spain) 
E-mail: pilar.abad@urjc.es 
 
Sonia Benito 
Universidad Nacional de Educación a Distancia (UNED) 
Senda del Rey 11  
28223, Madrid, Spain 
E-mail: soniabm@cee.uned.es 
 
Miguel Angel Sánchez Granero 
Universidad de Almería 
Crta. Sacramento s/n 
Almería, Spain 
E-mail: misanche@ual.es 
 
Carmen López 
Universidad Nacional de Educación a Distancia (UNED) 
 
 
* This work has been funded by the Spanish Ministerio de Ciencia y Tecnología (ECO2009-
10398/ECON and ECO2011-23959).  
 
 

Executive summary:  

This paper evaluates the performance of several skewed and symmetric distributions in 

modeling the tail behavior of daily returns and forecasting Value at Risk (VaR). First, we used 

some goodness of fit tests to analyze which distribution best fits the data. The comparisons in 

terms of VaR have been carried out examining the accuracy of the VaR estimate and minimizing 

the loss function from the point of view of the regulator and the firm. The results show that the 

skewed distributions outperform the normal and Student-t (ST) distribution in fitting portfolio 

returns. Following a two-stage selection process, whereby we initially ensure that the distributions 

provide accurate VaR estimates and then, focusing on the firm´s loss function, we can conclude 

that skewed distributions outperform the normal and ST distribution in forecasting VaR. From the 

point of view of the regulator, the superiority of the skewed distributions related to ST is not so 

evident. As the firms are free to choose the VaR model they use to forecast VaR, in practice, 

skewed distributions will be more frequently used.  

 
 
Keywords: Value at Risk, Parametric model, Skewness t-Generalised Distribution, GARCH 
Model, Risk Management, Loss function. 



 

 

1. Introduction 

A primary tool for financial risk assessment is Value at Risk (VaR). It is defined as the 

maximum loss expected of a portfolio of assets over a certain holding period at a given confidence 

level (probability). Since the Basel Committee on Bank Supervision at the Bank for International 

Settlements requires the financial institution to meet capital requirements on the basis of VaR 

estimates, allowing them to use internal models for VaR calculations, this measurement has 

become a basic market risk management tool for financial institutions.  

Despite VaR´s conceptual simplicity, its calculation could be rather complex. Many 

approaches have been developed to forecast VaR: non parametric approaches, e.g. Historical 

Simulation; semi-parametrics approaches, e.g. Extreme Value Theory and the Dynamic quantile 

regression CaViar model (Engle and Manganelli (2004)); and parametric approaches e.g. 

Riskmetrics (J.P. Morgan (1996)).  

The parametric approach is one of the most used by financial institutions. This approach 

usually assumes that the asset returns follow a normal distribution. This assumption simplifies the 

computation of VaR considerably. However, it is inconsistent with the empirical evidence of asset 

returns, which finds that the distribution of asset returns is skewed, fat-tailed, and peaked around 

the mean (see Bollerslev (1987)). This implies that extreme events are much more likely to occur 

in practice than would be predicted by the symmetric thinner-tailed normal distribution. 

Furthermore, the normality assumption can produce VaR estimates that are inappropriate measures 

of the true risk faced by financial institutions.  

Since the ST distribution has fatter tails than the normal one, this distribution has been 

commonly used in finance and risk management, particularly to model conditional asset returns 

(Bollerslev (1987)). The empirical evidence of this distribution performance in estimating VaR is 

ambiguous. Some papers show that the ST distribution performs better than the normal distribution 

(see Abad and Benito (2013), Orhan and Köksal (2012) and Polanski and Stoja (2010)) while other 

papers report that the ST distribution overestimates the proportion of exceptions (see Angelidis et 

al. (2007) and Guermat and Harris (2002)).  

The ST distribution can often account well for the excess kurtosis found in common asset 

returns, but this distribution does not capture the skewness of the returns. Taking this into account, 

one direction for research in risk management involves searching for other distribution functions 

that capture this characteristic. The skewness Student-t distribution (SSD) of Hansen (1994), the 

exponential generalized beta of the second kind (EGB2) of McDonald and Xu (1995), the 

generalised error distribution (GED) of Nelson (1991), the skewness generalised-t distribution 

(SGT) of Theodossiou (1998), the skewness error generalised distribution (SGED) of Theodossiou 

(2001) and the inverse hyperbolic sign (IHS) of Johnson (1949) are the most used in VaR 

literature. Some applications of skewness distributions to forecast the VaR can be found in Chen et 

al. (2012), Polanski and Stoja (2010), Bali and Theodossiou (2008), Bali et al. (2008), Haas et al. 



(2004), Zhang and Cheng (2005), Haas (2009), Ausín and Galeano (2007), Xu and Wirjanto 

(2010) and Kuester et al. (2006). Chen et al. (2012) compared the ability to forecast the VaR of a 

normal, ST, SSD and GED. In this comparison the SSD and GED distributions provide the best 

results. Polanski and Stoja (2010) compared the normal, ST, SGT and EGB2 distributions and 

found that just the latter two distributions provide accurate VaR estimates. Bali and Theodossiou 

(2008) compared a normal distribution with the SGT distribution and showed that the SGT 

provided a more accurate VaR estimate.  

In this paper we carry out a comprehensive comparison of the skewed distributions 

aforementioned: SSD, SGT, SGED and IHS. Besides, in this comparison we include both the 

normal and the ST distribution. The comparative is performed following two directions. First, we 

compare the distributions in statistical terms to determine which is the best for fitting financial 

returns. Then, we compare the distributions in terms of VaR, in order to select which is best for 

forecasting VaR.  

The main differences with the previous literature are as follows: (1) we consider a larger 

number of skewed distributions; (2) the comparison in statistical terms is made using a large 

battery of tests: Likelihood ratio, Chi-square (Chi2) of Pearson (1900) and Kolmogorov-Smirnov 

(KS) test (Kolmogorov (1933), Smirnov (1939) and Massey (1951)); the papers aforementioned 

only used the likelihood ratio test; 3) to carry out the comparison in terms of VaR we evaluate the 

results on the basis of two criteria: (i) the accuracy of VaR and (ii) the minimization of two loss 

functions which reflect the concerns of the  financial regulator and the firm (Sarma et al. (2003)).   

In the next section, we present the methodology used to estimate the VaR and summarize 

the statistical tests and the loss functions that we have used to evaluate the VaR estimates. In 

section 3, we present the data. The results of the comparison in statistical terms and in terms of 

VaR are presented in sections 4 and 5 respectively. The last section includes the main conclusions. 

2. Methodology 

According to Jorion (2001), VaR measure is defined as the worst expected loss over a 

given horizon under normal market conditions at a given level of confidence. The VaR is thus a 

conditional quantile of the asset return distribution. Let n1 2 3r , r , r ,..., r  be identically distributed 

independent random variables representing the financial returns. Use )(rF  to denote the 

cumulative distribution function,
1( ) Pr( )t tF r r r −= < Ω , conditionally on the information set 1t−Ω  

that is available at time t-1. Assume that { }tr  follows the stochastic process t tr µ ε= +  

where ( )01t t t tz z iid ,ε σ= ∼ , µ  is the conditional mean, tσ  the conditional standard deviation of 

returns. The VaR with a given probability ( )0 1,α ∈ , denoted by VaR( )α , is defined as the α  

quantile of the probability distribution of financial returns: tF(VaR( )) Pr( r VaR )( )α α α= < =  

Under the framework of the parametric techniques (see Jorion (2001)), the conditional VaR 

estimate can be calculated as ˆt t tVaR kαµ σ= + , where tµ  represents the conditional mean, which we 



assume is zero, ̂tσ  sigma is the conditional standard deviation and kα  denotes the corresponding 

quantile of the distribution of the standardized returns at a given confidence level 1-α .1  

Having obtained significant evidence from the Engle and Ng (1993) test on the fact that 

good and bad news have a different impact on conditional volatilities of asset returns, we use the 

Exponential GARCH model of Nelson (1991) to estimate tσ  needed for conditional VaR 

analysis2. Finally, once the variance has been calculated we estimate the distributions of the 

standardized returns under each of the considered distribution functions: normal, ST, SGT, SGED, 

SSD and IHS. Table 1 shows the density function of these skewed distributions.   

In the first stage, before the calculation of the VaR, we compare the distributions in 

statistical terms. To do this, we use a likelihood test (to compare the fit of two models) and two 

goodness of fit tests KS and Chi2 (to determine whether a sample can be considered as a draw 

sample from a given specified distribution). The KS test is based on the maximum difference 

between an empirical and a hypothetical cumulative distribution function. The Chi2 test is based 

on the probability distribution function and performs by grouping the data into bins, calculating the 

observed and expected counts for those bins.                                                                                             

In the second stage, we calculate the VaR and test the accuracy of the VaR estimate under 

these distributions. We use four standard tests: unconditional and conditional coverage tests, the 

Back-Testing criterion and the dynamic quantile test. We have an exception when 1tr +  < VaR( )α  

and then the exception indicator variable (It+1) is equal one (zero in other cases).  

Kupiec (1995) shows that the unconditional coverage test has as a null hypothesisα α⌢ = , 

with a likelihood ratio statistic ( ( )( ) ( )( )N x N xx x
UCLR log 1 log 1α α α α− − 

  
= 2 − − −⌢ ⌢ ), which follows an 

asymptotic 2(1)χ  distribution. A similar test for the significance of the deviation of α⌢  from α  is 

the back-testing criterion statistic ( ) ( )1Z N N Nα α α α= − / −⌢

 
which follows an asymptotic N (0,1) 

distribution. The conditional coverage test (Christoffersen (1998)) jointly examines if the 

percentage of exceptions is statistically equal to the expected one and the serial independence of 

It+1. The likelihood ratio statistic of the conditional coverage test is LRcc=LRuc+LRind, which is 

asymptotically distributed 2(2)χ , and the LRind statistic is the likelihood ratio statistic for the 

hypothesis of serial independence against first-order Markov dependence. Finally, the dynamic 

quantile test proposed by Engle and Manganelli (2004) examines if the exception indicator is 

uncorrelated with any variable that belongs to the information set 1t−Ω  available when the VaR 

was calculated. This is a Wald test of the hypothesis that all slopes are zero in a regression of the 

exception indicator variable on a constant, 5 lags and the VaR.  

Additionally, we evaluate the magnitude of the losses experienced. The model that 

minimizes the total loss is preferred to the other models. For this purpose, we have considered two 

                                                 
1 In case of the skewed distributions the kα value is a function of the skewness and kurtosis parameters. 
2 The EGARCH models have been estimated below a ST distribution. 



loss functions: the regulator loss function and the firm’s loss function. Lopez (1998, 1999) 

proposed a loss function, which reflects the utility function of a regulator. In this specification, the 

magnitude loss function assigns a quadratic specification when the observed portfolio losses 

exceed the VaR estimate. Thus, we penalize only when an exception occurs according to the 

following quadratic specification:  

( )2



<t t t t
t

VaR r if r VaRRLF
0 otherwise

-
=   (1) 

This loss function gives higher scores when failures take place and considers the magnitude 

of the failure. In addition, the quadratic term ensures that large failures are penalized more than 

small failures. 

But firms use VaR in internal risk management and, in this case, there is a conflict between 

the goal of safety and the goal of profit maximization. A too high VaR forces the firm to hold too 

much capital, imposing the opportunity cost of capital upon the firm. Taking this into account, 

Sarma et al. (2003) define the firm’s loss function as follows: 

( )2

.





<t t t t

t
t

VaR r if r VaRFLF
VaR otherwiseβ

-
=

-

   (2) 

β being the opportunity cost of capital.  

3. Data 

The data consist of closing daily returns on nine composite indexes from 1/1/2000 to 

11/30/2012 (around 3250 observations). The indexes are: Japanese Nikkei, Hong Kong Hang 

Seng, Israeli Tel Aviv (100), Argentine Merval, US S&P 500 and Dow Jones, UK FTSE100, the 

French CAC40 and the Spanish IBEX-35. The data were extracted from the Bloomberg database. 

The computation of the indexes returns (r t) is based on the formula, r t=ln(It)-ln(It-1) where It is the 

value of the stock market index for period t.  

Figure 1 shows the daily returns and Table 2 provides basic descriptive statistics of the 

data. For each index, the unconditional mean of daily return is very close to zero. The 

unconditional standard deviation is especially high for Merval (2.14). For the rest of stock index 

returns the standard deviation moves between 1.27 Dow Jones and 1.63 Hang Seng. Going back to 

Figure 1, we can see that the range fluctuation of the returns is not constant, which means that the 

variance of these returns changes over time.  

In order to gain some intuition, we adopt the volatility measure proposed by Franses and 

van Dijk (2000), wherein the volatility of returns is defined as: 

( )
2

2
1t t t tV r E r 

 
 

Ω
-

= -   (3) 

where 1tΩ
-

 is the information set at time t-1. Figure 2 presents tV  as “volatilities”. The volatility of 

the series was high during the early 2000s, especially in the Merval index. From 2001 to 2002 the 



conditional volatility of MERVAL was almost 1 point higher than the whole period, even greater 

than those showed from 2008 to 2009.  This corresponds to the Argentine economic crisis (1999–

2002) which was the major downturn in Argentine´s economy3. The period from 2003 to early 

2007 was very quiet. In August 2007 the financial market tensions started and they were followed 

by a global financial and economic crisis leading to a significant rise in the volatility of returns. 

This increase was especially important after August 2008 coinciding with the fall of Lehman 

Brothers. From 2008 to 2009, the volatility of the S&P500, Nikkei and IBEX35, measured by the 

standard deviation of returns was 2.42, 2.20, and 2.10 respectively. In the case of S&P500, the 

standard deviation was almost 1 point higher than the standard deviation of the whole period 2000-

2012 (1.57). A similar increase is observed in all indexes. In the last two years of the sample, we 

observe a period that is more stable than during the financial crisis. 

The skewness statistic is negative and significant for all the indexes considered except in 

the case of the CAC40 and the IBEX35. This means that the distribution of those returns is skewed 

to the left. When considering the CAC40 and the IBEX35 the skewness statistic is positive, 

implying that these distributions are skewed to the right but but jonly in the case of IBEX35 this 

statistic is statistically significant at 1% level.  

For all the indexes considered, the excess kurtosis statistic is very large and significant at 

1% level implying that the distributions of those returns have much thicker tails than the normal 

distribution. Similarly, the Jarque-Bera statistic is statistically significant rejecting the assumption 

of normality. These results are in line with those obtained by Bollerslev (1987), Bali and 

Theodossiou (2007), and Bali et al (2008), among others. All of them find evidence that the 

empirical distribution of the financial return is asymmetric and exhibits a significantly excess of 

kurtosis (fat tails and peakness).  

In order to capture the non-normal characteristics observed in our data set, we fit several 

skewed distributions: SGT, SGED, SSD and IHS. In this comparison we also include the normal 

and symmetric ST distributions. In Table 3 we present the estimated parameters of these 

distributions. This Table provides the estimates for the mean (µ) and the standard deviation (σ) of 

log-returns and its standard errors in brackets. As expected, these estimates are quite similar across 

distributions and do not differ much from the simple arithmetic means and standard deviations of 

log-returns presented in Table 2. The unconditional mean is close to zero for all the indexes and 

the unconditional standard deviation moves around 1.5 (in percentage) except Merval (2.14). As 

expected from the previous analysis, the Merval index is the most volatile index.  

The skewness parameter λ, for all indexes considered is negative and significant at the 1% 

level, which means that the distributions of these returns are skewed to the left. This result is in 

opposition to the preliminary evidence that suggested a symmetric distribution for CAC40 and a 

skewed distribution to the right for IBEX35.  

                                                 
3It began in 1999 with a decrease of the real Gross Domestic Product (GDP). The crisis caused the fall of the 
government, default on the country's foreign debt, widespread unemployment, riots, the rise of alternative currencies 
and the end of the peso's fixed exchange rate to the US dollar. 



On the other hand, the kurtosis parameters η and κ, in the case of SGT, the parameter κ 

controls mainly the peakness of the distribution around the mode, while the parameter η controls 

mainly the tails of the distribution (adjusting the tails to the extreme values). The parameter η has 

the degrees of freedom interpretation as in ST. For all the series and all distributions considered, 

the kurtosis parameters are highly significant. For the SGT, the value of κ is around 1.5, except for 

Nikkei and Tel Aviv which are 1.89 and 1.78 respectively. The value of η is around 4.5 for Nikkei, 

Merval, DJ, FTSE and CAC40. For the rest of the indexes it is a little bit higher. These estimates 

are quite different from those of the normal distribution (κ = 2 and η = ∞), which indicates that this 

set of returns is characterized by excess kurtosis.  

4. Comparison of the distributions in statistical terms 

In this section we want to answer the following question: Which distribution is the best one 

for fitting asset returns? 

The above results provide strong support to the hypothesis that stock returns are not 

normal. As the normal distribution is nested within the SGT, SGED and SSD distributions we can 

use the log-likelihood ratio for testing the null hypothesis of normality against that of SGT, SGED 

or SSD. For all the indexes considered, this statistic is quite large and statistically significant at the 

1% level, providing evidence against the normality hypothesis (see Table 4). Additional evidence 

against the normality hypothesis can be found in Figure 3 where we present the histogram and the 

density functions of several skewed distributions for the Nikkei index. We can see that all of these 

distributions provide a better fit than the normal ones4.  

To evaluate which is the most adequate, we perform several kinds of tests. First, as the SGT 

nets all the distributions considered in this paper (except IHS), we use the likelihood ratio test to 

evaluate which distribution is best for fitting the data5. Overall, for all the indexes considered, the 

likelihood statistics indicate rejection of the SGED, the SSD, and the ST in favour of the SGT (see 

Table 4). As the IHS is not nested in the SGT distribution, we cannot conclude that the SGT 

distribution is the best. So, to ensure the robustness of the results several alternative tests have been 

used: Chi2 and KS tests. Unlike the likelihood ratio test used to compare two distributions, the 

Chi2 and the KS tests are used to examine if the asset returns’ empirical distribution follows a 

particular theoretical distribution. The theoretical distributions we have considered are: normal, 

ST, SGT, SSD, SGED and IHS. The Chi2 statistic (see Table 4) suggests that the empirical 

distributions of the returns considered in this paper can be adequately characterized using two of 

the distributions we have considered: SGT and IHS. Both distributions seem to fit the data well in 

8 of the 9 indexes considered. For the Hang Seng, Tel Aviv and S&P 500 indexes, the SGED 

distribution cannot be refused either. On the other hand, the ST and the normal distributions do not 

fit any index. The KS test provides similar results (see Table 4). According to this test, the 
                                                 
4  The qualitative results of the remaining indexes are similar. We only represent the results for one index in order to 
free space. 
5 Specifically, it gives for η = ∞ the SGED, for κ = 2 the SSD, for λ=0 and κ = 2 the ST and for λ=0, n = ∞ and k = 2 
the normal distribution (see Hansen, McDonald and Theodossiou (2001) for a comprehensive survey on the skewed 
fat-tailed distributions). 



empirical distribution of all the indexes considered (except Nikkei) follows a SGT distribution. 

The IHS fits the data well in only five of the indexes (Merval, CAC40, IBEX35, Tel Aviv and 

Nikkei). According to this test, the SSD distribution fits the data well in four of the considered 

indexes (Merval, CAC40, IBEX35 and FTSIE) and the SGED distribution fits the data well in four 

indexes (Nikkei, Merval, IBEX35 and Hang Seng). The ST distribution only fits well in three of 

the nine indexes while the normal distribution does not do well in any index. 

Taking into account the results described in this section, we can conclude that the 

symmetric distributions (normal and ST) do not fit financial returns well. This is in line with the 

previous results shown in the above sections. Among the set of skewed distributions considered in 

this paper, the SGT distribution seems to be the best in fitting the data, followed closely by the IHS 

distribution. 

5 Evaluating the performance in terms of VaR  

In this section we compare the normal, the ST and the skewed distributions in terms of 

VaR. The comparison is carried out evaluating (i) the accuracy of the VaR estimates and (ii) the 

losses that VaR produces. For each distribution, we use parametric approaches to forecast the VaR 

out-of-the-sample one-step-ahead at 1% and 0.25% confidence level. The analysis period runs 

from the first of January 2008 to the end of December 2009.  We choose this period because it is 

characterized by a high volatility all over the world so that it is known in financial literature as the 

Financial Global Crisis period. In Figure 1, we highlight in black the period analyzed.  

5.1 Back Testing 

The results of the accuracy test are presented in Tables 5 and 6. In Table 5 we show the 

results of the accuracy test at 1% confidence level and Table 6 reports the results at 0.25% 

confidence level. In both tables, we present the percentage of exceptions obtained with each 

distribution: normal, ST, SSD, SGED, SGT and IHS together with Riskmetric. Below these 

percentages, we present the five statistics used to test the accuracy of the VaR estimates. When the 

null hypothesis that “the VaR estimate is accurate” has not been rejected by any test, we have 

shaded the area (the percentage of exceptions).  

In the analyzed period the VaR estimates obtained under a normal distribution are very 

poor. For almost all the indexes considered, the parametric approach under a normal distribution 

underestimates risk at the 1% and 0.25% confidence levels. This result does not depend on the 

volatility model we have used to forecast the VaR, EGARCH or MME (Riskmetrics). 

At the 1% confidence level, the VaR estimate provided by the skewed distributions and the 

ST distribution is quite accurate. At this confidence level, the SGT and the HIS perform well in 

eight of the nine indexes considered, only failing in the IBEX35. The ST, the SSD and the SGED 

distributions provide accurate VaR estimates in seven of the 9 cases considered. At the 0.25% 

confidence level, all the skewed distributions provide accurate VaR estimates in eight of the nine 

indexes considered, except the IHS that fails in two cases. At this confidence level, the ST 



distribution performs well in five of the nine indexes considered: Nikkei, S&P500, DJ, CAC40 and 

IBEX35. In the case of Merval, Hang Seng and Tel Aviv, this distributions overestimate risk. 

Then, at the higher confidence level the evidence in favor of the skewed distributions related to the 

ST one is more obvious.  

5.2 Loss Functions 

In this section we evaluate the VaR estimate in terms of the regulator loss function (Table 

7) and the firm’s loss function (Table 8). The results in Table 7 have been multiplied by 1000 

given the small value obtained. The data marked in bold type represents the minimum value for 

this function in each case.  

From the regulator loss function (see Table 7), we find that the parametric approach under a 

normal distribution joined to Riskmetrics provide the highest losses while the ST distribution 

provides the lowest losses followed by the IHS and the SGT distributions. Among the skewed 

distributions, the SSD gives the worst outcome in all cases. According to this result, we can 

conclude that from the point of view of the regulator the best distribution is the ST, as this 

distribution is the most conservative.  

The problem associated with the regulator loss function is that this function does not take 

into account the firms’ opportunity cost. So that one model that overestimates the risk, as the ST 

distribution does in three of the cases, may be considered the most appropriate. Taking this into 

account we calculate the losses from a firm´s point of view.6   

In terms of the firm’s loss function (see Table 8), the normal distribution provides the 

lowest losses while the ST distribution shows the highest losses. This result is coherent since it is 

well known that the normal distribution underestimates risk providing the lowest capital 

opportunity cost. Since the ST distribution tends to overestimate risk, the capital opportunity cost 

with this distribution is the highest. The magnitudes of losses obtained by all the skewed 

distribution are very similar. In terms of this loss function, the best skewed distribution is the SSD. 

This distribution obtains the lowest losses in seven of the nine cases. The SGT distribution, 

although it is not the best, works out well giving lower losses than the ST does.  

On the whole, following this selection process in two stages, where first we ensure that the 

distributions provide accurate VaR estimate and then focusing in the firm’s loss function, we can 

conclude that the skewed and fat tail distributions outperformed the normal and the ST 

distribution. From a point of view of the regulator, the superiority of the skewed distributions 

related to the ST is not so clear. 

 

6. Conclusion 

                                                 
6 In order to calculate the firm’s loss function we need to know the cost of capital. For this purpose, we have used the 
daily data of the interest rate of the Eurosystem monetary policy operations for the European indexes. For the rest of 
the indexes, we took the interest rate of the open market operations used by the Federal Reserve in the implementation 
of its monetary policy. 



This paper evaluates the performance of several skewed and symmetric distributions in 

modeling the tail behavior of daily returns and in forecasting VaR. The skewed distributions 

considered are: (i) the skewed Student-t distribution of Hansen (1994); (ii) the skewed error 

generalised distribution of Theodossiou (2001); (iii) the skewed generalised-t distribution of 

Theodossiou (1998) and (iv) the inverse hyperbolic sign of Johnson (1949). The symmetric 

distributions are the normal and the Student-t ones.  

For this study we have used daily returns on nine composite indexes: the Japanese Nikkei, 

Hong Kong Hang Seng, Israeli Tel Aviv (100), Argentine Merval, US S&P 500 and Dow Jones, 

UK’s FTSE100, the French CAC40 and the Spanish IBEX-35. The sample used for the statistical 

analysis runs from January 2000 to the end of November 2012. The analysis period for forecasting 

VaR runs from 2008 to 2009, which is known as the Global Financial Crisis period.  

From the results presented in the paper, we can conclude that the skewness and fat tail 

distributions outperform the normal one in fitting financial returns and forecasting VaR. Among all 

the skewed distributions considered in this paper, the skewed generalised-t distribution of 

Theodossiou (1998) is the best one in fitting data. However, in terms of their ability to forecast the 

VaR, we do not find significant differences as all of them provide accurate VaR estimates for a 

high number of indexes and produce similar losses.  

Finally, we find evidence in favor of the skewed distributions compared to the ST 

distribution.  In statistical terms, most of them fit the data better than the ST. In terms of value at 

risk, the accuracy VaR test indicates that the skewed distributions outperform the ST. On the other 

hand, with regards to the loss function, the result depends on the kind of function we use to 

measure the losses. From a point of view of the regulator, ST distribution is the best in forecasting 

VaR as this distribution provides the more conservative VaR estimate. However, from the point of 

view of the firm, the skewed distributions outperform the ST distribution, since the latter 

distribution tends to raise the firm´s capital cost. As companies are free to choose the VaR model 

they use to forecast VaR, it is clear that they will prefer the skewed distributions. 

 



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12 

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13 

Table 1. Density functions 
  

Formulations 
 

Restrictions 

SSD of Hansen 
(1994) 

1 2
2

1 2
2

11
2 1

11
2 1

( ) /

t
t

t ( ) /

t
t

b z ab c if z
b

f ( z v , )

b z ab c if z
b

η

η

α
η η

η
α

η η

− +

− +


                   


  

    
         

  

++ < −− −
=

++ ≥ −− +

 2 2 224 1 3
1

1
2

2
2

t t t t  z = ( r  - µ ) /

a c b a

c

)

σ

ηλ λη
ηΓ

ηπ η Γ

 
 
 
 

 
  
 

  
  
   

−= = + −−
+

=
−

 

1

2

λ
η

<
>

SGED of  
Theodossiou 

(2001) 
( )1

k
t

t k k
t

zCf ( z ,k ) exp
sign( z )

δ
λ σ δ λ θ

 
 
 
 
 
 

+
= −

+ +
 

 

t t t t  z = ( r  - µ ) / σ  
( ) 1

0 5 0 5 1

1 2 2 2

2 1 2

1 3

2 1 3 4

. .

C k / ( / k ) AS( )

( / k ) ( / k ) S( )

AS( ) S( ) A

θ Γ δ λ λ
θ Γ Γ λ

δ λ λ λ λ λ

−

− −

−

= =

=
= = + −  

1   skewed parameter

k=kurtosis parameter

λ <

 

SGT of 
Theodossiou 

(1998) 
( ) ( )

1

1
1 1

kk
t

t k k
t

z
f ( z , ,k ) C

/ k s i g n ( z )

η

δ
λ η

η δ λ θ

+−
 
 
 
  
     

+
= +

+ + +
 

 

1 1
1

2

11

11
2

1 1 10 5

1 1 1 22

1 1 1 31 3

k

k

k

C , k B ,
k k k g

B , B ,
k k k k k

g ( )B , B ,
k k k k k

η η θ θ
ρ

η η ηρ λ

η η ηλ δ ρθ

− −
−

−

−

   
      
   

     
          
     

     
          
     

+= =
−

+ −=

+ −= + =

 

1    s k e w e d  p a r a m e t e rλ <

 
2 t a i l - t h i c k n e s s  p a r a m e t e r η >  

k > 0  p e a k e d n e s s  p a r a m e te r

 
 P ea rso n 's  skew nessδ  

t t t t  z = ( r  - µ ) / σ

IHS of Johnson 
(1949) ( )

( ) ( )
2

2 22

22 2
2

t t t

t

k kIHS( z ,k ) exp ln z z ( ln( )
z

λ δ θ δ λ θ
π θ δ

 
   
            

 

= − × − + + + + − +
+ +

 

 
1 w/θ σ=   

w w/δ µ σ=  
2 2 22 2 0 50 5 2 1k k . k

w . (e e ) (e )λ λσ − − −+ − += + + −

w

w

 mean

 standard deviation

w  

x standard normal variable

sinh( x / k )

µ
σ

λ= +
 

Note: In all these distributions z represents the standardized returns.  



 

 

14 

 

Table 2. Descriptive Statistics 
 

 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque Bera 

Nikkei -0.022 0.004 13.234 -12.111 1.568 
-0.393** 
(0.044) 

9.686** 
(0.087) 

5996 
(0.001) 

Hang Seng 0.008 0.044 13.407 -13.582 1.632 
-0.065 
(0.043) 

10.386** 
(0.087) 

7253 
(0.001) 

Tel Aviv 0.024 0.055 9.782 -8.425 1.338 
-0.311** 
(0.044) 

6.945** 
(0.087) 

2107 
(0.001) 

Merval 0.047 0.090 16.117 -12.952 2.140 
-0.093* 
(0.043) 

7.944** 
(0.087) 

3243 
(0.001) 

S&P 500 -0.001 0.050 10.957 -9.47 1.354 
-0.158** 
(0.043) 

10.293** 
(0.086) 

7212 
(0.001) 

Dow Jones 0.010 0.049 10.089 -8.7 1.265 
-0.185** 
(0.043) 

9.372** 
(0.086) 

5515 
(0.001) 

Ftsie100 -0.004 0.025 9.384 -9.266 1.301 
-0.135** 
(0.043) 

8.692** 
(0.086) 

4416 
(0.001) 

CAC40 -0.015 0.019 10.595 -9.472 1.572 
0.038 

(0.043) 
7.494** 
(0.085) 

2782 
(0.001) 

IBEX35 -0.012 0.060 13.484 -9.5858 1.576 
0.1227** 
(0.043) 

7.8219** 
(0.086) 

3177 
(0.001) 

Note: This table presents the descriptive statistics of the daily percentage returns of Nikkei, Hang Seng, Tel Aviv 
100, Merval, S&P 500, Dow Jones, Ftsie 100, CAC-40 and IBEX-35. The sample period is from January 2nd, 2000 
to November 30th, 2012. The index return is calculated as Rt=100(ln(It)-ln(I t-1)) where It is the index level for period 
t. Standard errors of the skewness and excess  kurtosis are calculated as n/6    and n24  respectively. The JB 

statistic is distributed as the Chi-square with two degrees of freedom. *, ** denote significance at the 5% and 1% 
level respectively. 



 

 

15 

Table 3. Maximum likelihood estimates of alternative distribution functions 
Nikkei  µ S.E σ S.E λ S.E η S.E κ S.E 

SGT 0.000 (0.000) 0.016** (0.001) -0.047* (0.021) 4.766** (0.282) 1.896** (0.078) 
SGED 0.000 (0.000) 0.015** (0.000) -0.041** (0.004)   1.133** (0.033) 
SSD 0.000 (0.000) 0.016** (0.000) -0.048* (0.021) 4.442** (0.236)   
IHS 0.000 (0.000) 0.015** (0.000) -0.086 (0.032)   1.472** (0.054) 
ST 0.000 (0.000) 0.016** (0.001)   4.404** (0.232)   

Normal  0.000 (0.000) 0.016** (0.000)       
Hang Seng µ S.E σ S.E λ S.E η S.E κ S.E 

SGT 0.000 (0.000) 0.016** (0.001) -0.034** (0.014) 6.328** (0.547) 1.338** (0.044) 
SGED 0.000 (0.000) 0.016** (0.000) -0.031 (--)   0.977** (0.028) 
SSD 0.000 (0.000) 0.017** (0.000) -0.041* (0.018) 3.314** (0.100)   
IHS 0.000 (0.000) 0.016** (0.000) -0.067* (0.027)   1.21 (0.033) 
ST 0.000 (0.000) 0.017** (0.001)   3.297** (0.100)   

Normal  0.000 (0.000) 0.016** (0.000)       
Tel Aviv µ S.E σ S.E λ S.E η S.E κ S.E 

SGT 0.000 (0.000) 0.013** (0.001) -0.060** (0.021) 5.247** (0.365) 1.785** (0.068) 
SGED 0.000 (0.000) 0.013** (0.000) -0.052** (0.016)   1.175** (0.035) 
SSD 0.000 (0.000) 0.014** (0.000) -0.062** (0.021) 4.381** (0.232)   
IHS 0.000 (0.000) 0.013** (0.000) -0.102** (0.032)   1.463** (0.054) 
ST 0.001** (0.000) 0.014** (0.001)   4.331** (0.228)   

Normal 0.000 (0.000) 0.013** (0.000)       
Merval  µ S.E σ S.E λ S.E η S.E κ S.E 

SGT 0.000 (0.000) 0.022** (0.001) -0.043* (0.018) 4.456** (0.241) 1.531** (0.051) 
SGED 0.000 (0.000) 0.021** (0.000) -0.033** (0.002)   0.998** (0.028) 
SSD 0.000 (0.000) 0.023** (0.000) -0.047** (0.018) 3.083** (0.075)   
IHS 0.000 (0.000) 0.022** (0.000) -0.068* (0.027)   1.171** (0.029) 
ST 0.001* (0.000) 0.023** (0.001)   3.088** (0.078)   

Normal 0.000 (0.000) 0.021** (0.000)       
S&P 500 µ S.E σ S.E λ S.E η S.E κ S.E 

SGT 0.000 (0.000) 0.014** (0.001) -0.064** (0.013) 5.735** (0.430) 1.239** (0.038) 
SGED 0.000 (0.000) 0.013** (0.000) -0.062 (--)   0.902** (0.008) 
SSD 0.000 (0.000) 0.016** (0.000) -0.069** (0.016) 2.760** (0.046)   
IHS 0.000 (0.000) 0.014** (0.000) -0.087** (0.024)   1.079** (0.023) 
ST 0.000 (0.000) 0.015** (0.001)   2.770** (0.049)   

Normal 0.000 (0.000) 0.014** (0.000)       
Dow Jones µ S.E σ S.E λ S.E η S.E κ S.E 

SGT 0.000 (0.000) 0.013** (0.001) -0.058** (0.017) 4.496** (0.241) 1.524** (0.051) 
SGED 0.000 (0.000) 0.012** (0.000) -0.057** (0.002)   0.983** (0.027) 

SSD 0.000 (0.000) 0.014** (0.000) -0.059** (0.018) 3.122** (0.078)   
IHS 0.000 (0.000) 0.013** (0.000) -0.088** (0.026)   1.178** (0.029) 
ST 0.000 (0.000) 0.014** (0.001)   3.122** (0.080)   

Normal 0.000 (0.000) 0.013** (0.000)       
Ftsie100 µ S.E σ S.E λ S.E η S.E κ S.E 

SGT 0.000 (0.000) 0.013** (0.001) -0.054** (0.018) 4.273** (0.212) 1.623** (0.055) 
SGED 0.000 (0.000) 0.013** (0.000) -0.049** (0.003)   1.015** (0.028) 
SSD 0.000 (0.000) 0.014** (0.000) -0.056** (0.018) 3.237** (0.089)   
IHS 0.000 (0.000) 0.013** (0.000) -0.083** (0.027)   1.208** (0.031) 
ST 0.000 (0.000) 0.014** (0.001)   3.231** (0.091)   

Normal 0.000 (0.000) 0.013** (0.000)       

CAC40 µ S.E σ S.E λ S.E η S.E κ S.E 
SGT 0.000 (0.000) 0.016** (0.001) -0.062** (0.018) 4.545** (0.249) 1.673** (0.059) 

SGED 0.000 (0.000) 0.015** (0.000) -0.044* (0.021)   1.065** (0.030) 
SSD 0.000 (0.000) 0.016** (0.000) -0.066** (0.019) 3.540** (0.120)   
IHS 0.000 (0.000) 0.016** (0.000) -0.094** (0.028)   1.277** (0.036) 
ST 0.000 (0.000) 0.016** (0.001)   3.533** (0.122)   

Normal 0.000 (0.000) 0.016** (0.000)       

IBEX35 µ S.E σ S.E λ S.E η S.E κ S.E 
SGT 0.000 (0.000) 0.016** (0.001) -0.073** (0.017) 7.127** (0.717) 1.380** (0.045) 

SGED 0.000 (0.000) 0.016** (0.000) -0.068 (--)   1.050** (0.030) 
SSD 0.000 (0.000) 0.017** (0.000) -0.069** (0.018) 3.548** (0.125)   
IHS 0.000 (0.000) 0.016** (0.000) -0.092** (0.028)   1.270** (0.037) 
ST 0.000 (0.000) 0.016** (0.001)   3.584** (0.132)   

Normal 0.000 (0.000) 0.016** (0.000)       

Note: Parameter estimates of the Normal, SGT, SGED, SSD, IHS and  ST. S.E. denotes standard errors (in parentheses). Nine stock market 
returns in the period 1/1/2000-11/30/2012. µ, σ, λ and η are the estimated mean, standard deviation, skewness parameter, and tail-tickness 
parameter; к  represents the peakness parameter. An * (** ) denotes significance at the 5% (1%) level. 



 

 

16 

Table 4. Goodness-of-fit tests 

  Log-L LR_Normal LR_SGT Chi2 KS 
Nikkei                

SGT 8920.4 463.2** -- 5.239 (0.022)** 0.031 (0.004) 
SGED 8897.4 417.2** 46.0** 7.715 (0.006) 0.027 (0.021)** 
SSD 8920.3 463.0** 0.2 13.448 (0.001) 0.034 (0.001) 
IHS 8918.6 -- -- 3.453 (0.063)* 0.029 (0.011)** 
ST 8918.2 -- 4.4 20.958 (0.000) 0.029 (0.008) 

Normal 8688.8  -- 124.218 (0.000) 0.058 (0.000) 
Merval                

SGT 8016.9 612.6** -- 8.164 (0.017)** 0.019 (0.197)* 
SGED 8003 584.8** 27.8** 12.318 (0.002) 0.027 (0.021)** 
SSD 8012.5 603.8** 8.8* 15.965 (0.003) 0.020 (0.147)* 
IHS 8017 -- -- 6.005 (0.111)* 0.018 (0.260)* 
ST 8010.4 -- 13.0** 18.687 (0.000) 0.024 (0.053)* 

Normal 7710.6 -- -- 253.700 (0.000) 0.072 (0.000) 
S&P 500               

SGT 9777.7 824.2** -- 14.092 (0.001) 0.028 (0.013)* 
SGED 9769.2 807.2** 17.0** 8.761 (0.013)** 0.033 (0.002) 
SSD 9762.2 793.2** 31.0** 35.861 (0.000) 0.038 (0.000) 
IHS 9769.2 -- -- 22.316 (0.000) 0.035 (0.001) 
ST 9757.1 -- 41.2** 33.963 (0.000) 0.037 (0.000) 

Normal 9365.6 -- -- 266.854 (0.000) 0.080 (0.000) 
Dow Jones               

SGT 9929.7 682.6** -- 6.333 (0.042)** 0.028 (0.011)** 
SGED 9914.2 651.6** 31.0** 24.553 (0.000) 0.032 (0.002) 
SSD 9925.1 673.4** 9.2** 21.875 (0.000) 0.034 (0.001) 
IHS 9928.4 -- -- 8.647 (0.034)** 0.029 (0.007) 
ST 9921.6 -- 16..2** 30.360 (0.000) 0.030 (0.007) 

Normal 9588.4 -- -- 256.272 (0.000) 0.071 (0.000) 
CAC40               

SGT 9297.4 523.6** -- 3.209 (0.201)* 0.023 (0.067)* 
SGED 9281 490.8** 32.8** 17.858 (0.000) 0.033 (0.002) 
SSD 9295.3 519.4** 4.2* 7.248 (0.027)** 0.027 (0.018)** 
IHS 9297.4 -- -- 2.761 (0.430)* 0.022 (0.079)* 
ST 9291.1 -- 12.6** 38.232 (0.000) 0.025 (0.030)** 

Normal 9035.6 -- -- 191.314 (0.000) 0.064 (0.000) 
IBEX35               

SGT 9176.8 484.2** -- 3.767 (0.052)* 0.027 (0.018)** 
SGED 9169.8 470.2** 14.0** 11.509 (0.001) 0.028 (0.011)** 
SSD 9167.1 464.8** 19.4** 13.293 (0.001) 0.028 (0.011)** 
IHS 9170.9 -- -- 7.174 (0.067)* 0.029 (0.010)** 
ST 9162.4 -- 28.8** 25.413 (0.000) 0.034 (0.001) 

Normal 8934.7 -- -- 118.562 (0.000) 0.065 (0.000) 
Hang Seng               

SGT 8927.5 649.0** -- 1.543 (0.214)* 0.027 (0.020)** 
SGED 8918.4 630.8** 18.2** 5.519 (0.063)* 0.029 (0.010)** 
SSD 8916.3 626.6** 22.4** 9.290 (0.002) 0.037 (0.000) 
IHS 8920.4 -- -- 1.873 (0.392)* 0.034 (0.001) 
ST 8914.6 -- 25.8** 15.599 (0.000) 0.035 (0.001) 

Normal 8603 -- -- 23.434 (0.000) 0.072 (0.000) 
Tel Aviv               

SGT 9358.2 316.8** -- 5.721 (0.057)* 0.027 (0.023)** 
SGED 9343.6 332.6** 29.2** 4.288 (0.039)** 0.034 (0.002) 
SSD 9357.3 360.0** 1.8 11.097 (0.004) 0.029 (0.008) 
IHS 9358.6 -- -- 5.878 (0.053)* 0.026 (0.024)** 
ST 9354 -- 8.4* 33.459 (0.000) 0.025 (0.041)** 

Normal 9177.3 -- -- 106.813 (0.000) 0.058 (0.000) 
Ftsie100               

SGT 9857 628.2** -- 3.311 (0.191)* 0.025 (0.037)** 
SGED 9839.1 592.4** 35.8** 10.540 (0.005) 0.034 (0.001) 
SSD 9854.2 622.6** 5.6* 16.291 (0.000) 0.027 (0.018)** 
IHS 9857.3 -- -- 4.518 (0.211)* 0.027 (0.015)** 
ST 9851.2 -- 11.6** 25.173 (0.000) 0.029 (0.007) 

Normal 9542.9 -- -- 203.848 (0.000) 0.072 (0.000) 

Note: Log-L is the maximum likelihood value. LRNormal is the LR statistic from testing the null 
hypothesis that the daily returns are distributed as Normal against they are distributed as SGT, SGED or 
SSD. LRSGT is the LR statistic from testing the null hypothesis of alternative distribution against the 
SGT. Chi2 and KS denote Chi-square and Kolmogorov Smirnov tests. Figures in brackets denote p-
value. An *(* * ) denotes significance at the 5%(1%) level.  

 
 



 

 

17 

 
 
 
 

Table 5. Accuracy test, 1% level 
 

 Nikkei Merval S&P500 DJ CAC40 IBEX35 Hang Tel Aviv Ftsie100 

 

VaR_Normal 2.87% 2.24% 3.56% 2.77% 2.34% 2.17% 1.62% 2.64% 3.55% 

LRUC 4.970* 2.450 8.770** 4.696* 2.943 2.270 0.700 3.997* 8.725** 

BTC 4.149** 2.762** 5.792** 4.003** 3.056** 2.640** 1.384 3.653** 5.771** 

LRIND 0.310 0.219 0.579 0.348 0.251 0.212 1.105 0.388 0.577 

LRcc 5.280 2.670 9.349** 5.043 3.193 2.482 1.805 4.386 9.301** 

DQ 1.969 2.770 0.362 0.578 1.053 2.477 2.906 0.655 1.484 

VaR_MME 2.05% 2.85% 2.38% 1.58% 1.17% 1.77% 1.82% 2.23% 2.37% 

LRUC 1.808 4.920* 3.027 0.642 0.063 1.079 1.177 2.429 3.003 

BTC 2.329* 4.123** 3.108** 1.319 0.391 1.748 1.836 2.748** 3.093** 

LRIND 0.807 0.358 0.254 0.112 0.062 0.141 0.914 0.219 0.253 

LRcc 2.615 5.278 3.281 0.754 0.125 1.221 2.092 2.647 3.256 

DQ 5.132* 3.668 1.331 0.295 2.339 4.004* 2.289 3.758* 2.067 

VaR_T 1.64% 0.61% 1.19% 0.99% 1.17% 1.18% 0.61% 0.61% 2.17% 

LRUC 0.734 0.379 0.074 0.000 0.063 0.069 0.389 0.385 2.280 

BTC 1.420 -0.866 0.425 -0.022 0.391 0.410 -0.877 -0.874 2.647** 

LRIND 0.089 0.016 0.063 0.044 0.062 0.062 0.016 0.016 0.212 

LRcc 0.822 0.395 0.137 0.044 0.125 0.132 0.405 0.401 2.493 

DQ 3.652 0.047 2.423 0.253 0.145 9.879** 0.136 0.071 2.406 

VaR_SGT 1.84% 1.43% 1.78% 1.39% 1.17% 1.57% 1.01% 1.01% 1.78% 

LRUC 1.222 0.345 1.100 0.295 0.063 0.627 0.000 0.000 1.086 

BTC 1.874 0.948 1.767 0.872 0.391 1.302 0.027 0.032 1.754 

LRIND 0.116 0.088 0.142 0.086 0.062 0.111 0.045 0.045 0.142 

LRcc 1.338 0.433 1.242 0.381 0.125 0.738 0.045 0.045 1.228 

DQ 2.689 0.238 0.940 0.142 0.145 5.068* 3.156 0.185 0.229 

VaR_IHS 1.84% 1.43% 1.78% 1.19% 1.17% 1.57% 1.01% 1.01% 1.58% 

LRUC 1.222 0.345 1.100 0.074 0.063 0.627 0.000 0.000 0.632 

BTC 1.874 0.948 1.767 0.425 0.391 1.302 0.027 0.032 1.308 

LRIND 0.116 0.088 0.142 0.063 0.062 0.111 0.045 0.045 0.112 

LRcc 1.338 0.433 1.242 0.137 0.125 0.738 0.045 0.045 0.743 

DQ 2.688 0.237 0.942 0.121 0.145 5.069* 3.153 0.185 0.134 

VaR_SSD 1.84% 1.83% 2.18% 1.39% 1.17% 1.57% 1.21% 1.62% 1.97% 

LRUC 1.222 1.199 2.301 0.295 0.063 0.627 0.093 0.706 1.639 

BTC 1.874 1.855 2.661** 0.872 0.391 1.302 0.479 1.390 2.201* 

LRIND 0.116 0.146 0.213 0.086 0.062 0.111 0.064 0.115 0.175 

LRcc 1.338 1.346 2.514 0.380 0.125 0.738 0.158 0.821 1.814 

DQ 2.689 1.608 0.928 0.142 0.145 5.067* 2.134 0.254 0.824 

VaR_SGED 1.84% 1.43% 1.78% 1.39% 1.17% 1.57% 1.01% 1.22% 1.97% 

LRUC 1.222 0.345 1.100 0.295 0.063 0.627 0.000 0.095 1.639 

BTC 1.874 0.948 1.767 0.872 0.391 1.302 0.027 0.484 2.201* 

LRIND 0.116 0.088 0.142 0.086 0.062 0.111 0.045 0.064 0.175 

LRcc 1.338 0.433 1.242 0.381 0.125 0.738 0.045 0.160 1.814 

DQ 2.689 0.237 0.941 0.142 0.145 5.067* 3.156 0.067 0.825 

Note: The statistics are as follows: (i) the unconditional coverage test (LRuc); (ii) the back-testing criterion 
(BTC); (iii) statistics for serial independence (LRind); (iv) the Conditional Coverage test (LRcc) and (v) the 
Dynamic Quantile test (DQ). An **, (*) denotes rejection at 1% (5%) level. The shaded cells indicate that the 
null hypothesis that the VaR estimate is accurate is not rejected by any test. 



 

 

18 

 

 
 

Table 6. Accuracy test, 0.25% level 

 Nikkei Merval S&P 500 Dow Jones CAC40 IBEX35 Hang Seng Tel Aviv Ftsie100 

Panel A: 2008-09  

VaR_Normal 0.82% 0.81% 1.19% 0.59% 0.98% 1.18% 0.61% 0.61% 1.18% 

LRUC 1.718 1.703 4.028 0.749 2.698 4.003* 0.782 0.786 4.011* 

BTC 2.520* 2.506* 4.222** 1.548 3.292** 4.201** 1.590 1.594 4.209** 

LRIND 0.016 0.029 0.063 0.016 0.043 0.062 0.016 0.016 0.063 

LRcc 1.734 1.732 4.090 0.764 2.741 4.065 0.798 0.802 4.074 

DQ 0.023 0.044 0.080 0.089 0.080 9.833** 0.127 0.071 0.142 

VaR_MME 1.23% 1.63% 0.99% 0.40% 0.59% 0.59% 1.42% 0.61% 1.38% 

LRUC 4.170* 1.629 2.743 0.159 0.728 0.740 5.570* 0.786 5.439* 

BTC 4.333** 6.120** 3.331** 0.657 1.522 1.537 5.194** 1.594 5.098** 

LRIND 0.045 0.115 0.044 0.007 0.015 0.016 0.088 0.016 0.085 

LRcc 4.215 7.299* 2.787 0.166 0.743 0.755 5.658 0.802 5.525 

DQ 3.301 1.100 0.275 0.037 10.336** 10.309** 1.484 0.329 6.901** 

VaR_ST 0.20% 0.00% 0.40% 0.20% 0.59% 0.20% 0.00% 0.00% 0.99% 

LRUC 0.018 1.068 0.159 0.026 0.728 0.027 1.074 1.072 2.730 

BTC -0.199 -1.109 0.657 -0.234 1.522 -0.240 -1.113 -1.112 3.320** 

LRIND 0.002 NaN 0.007 0.002 0.015 0.002 NaN NaN 0.043 

LRcc 0.020 NaN 0.166 0.027 0.743 0.029 NaN NaN 2.774 

DQ 0.163 NaN 0.038 0.131 0.016 0.034 NaN NaN 0.077 

VaR_SGT 0.20% 0.20% 0.59% 0.40% 0.59% 0.20% 0.40% 0.20% 0.99% 

LRUC 0.018 0.020 0.749 0.159 0.728 0.027 0.174 0.020 2.730 

BTC -0.199 -0.206 1.548 0.657 1.522 -0.240 0.689 -0.210 3.320** 

LRIND 0.002 0.002 0.016 0.007 0.015 0.002 0.007 0.002 0.043 

LRcc 0.020 0.021 0.764 0.166 0.743 0.029 0.181 0.022 2.774 

DQ 0.166 0.102 0.059 0.036 0.015 0.027 0.027 0.013 0.073 

VaR_IHS 0.20% 0.20% 0.59% 0.40% 0.59% 0.20% 0.00% 0.20% 0.79% 

LRUC 0.018 0.020 0.749 0.159 0.728 0.027 1.074 0.020 1.626 

BTC -0.199 -0.206 1.548 0.657 1.522 -0.240 -1.113 -0.210 2.430* 

LRIND 0.002 0.002 0.016 0.007 0.015 0.002 NaN 0.002 0.028 

LRcc 0.020 0.021 0.764 0.166 0.743 0.029 NaN 0.022 1.654 

DQ 0.162 0.100 0.061 0.036 0.015 0.027 NaN 0.012 0.068 

VaR_SSD 0.20% 0.61% 0.59% 0.40% 0.59% 0.20% 0.40% 0.41% 0.99% 

LRUC 0.018 0.792 0.749 0.159 0.728 0.027 0.173 0.175 2.730 

BTC -0.199 1.602 1.548 0.657 1.522 -0.240 0.689 0.692 3.319** 

LRIND 0.002 0.016 0.016 0.007 0.015 0.002 0.007 0.007 0.043 

LRcc 0.020 0.808 0.764 0.166 0.743 0.029 0.181 0.182 2.774 

DQ 0.169 0.050 0.058 0.036 0.015 0.024 0.025 0.043 0.072 

VaR_SGED 0.20% 0.41% 0.59% 0.40% 0.59% 0.20% 0.40% 0.20% 0.99% 

LRUC 0.018 0.178 0.749 0.159 0.728 0.027 0.174 0.020 2.730 

BTC -0.199 0.698 1.548 0.657 1.522 -0.240 0.689 -0.210 3.320** 

LRIND 0.002 0.007 0.016 0.007 0.015 0.002 0.007 0.002 0.043 

LRcc 0.020 0.185 0.764 0.166 0.743 0.029 0.181 0.022 2.774 

DQ 0.169 0.135 0.058 0.036 0.015 0.024 0.027 0.012 0.073 

Note: The statistics are as follows: (i) the unconditional coverage test (LRuc); (ii) the back-testing criterion 
(BTC); (iii) statistics for serial independence (LRind); (iv) the Conditional Coverage test (LRcc) and (v) the 
Dynamic Quantile test (DQ). An **, (*) denotes rejection at 1% (5%) level. The shaded cells indicate that the 
null hypothesis that the VaR estimate is accurate is not rejected by any test. 



 

 

19 

 

 
 

Table 7. Magnitude of the regulatory loss function  

 level NORMAL MME ST SGT IHS SSD SGED 

Nikkei 
1.00% 0.00338 0.00860 0.00134 0.00186 0.00176 0.00212 0.00186 

0.25% 0.00065 0.00397 0.00004 0.00015 0.00008 0.00020 0.00015 

Merval 
1.00% 0.00667 0.00833 0.00053 0.00256 0.00244 0.00340 0.00251 

0.25% 0.00191 0.00307 0.00000 0.00013 0.00009 0.00039 0.00022 

S&P 500 
1.00% 0.00617 0.00343 0.00337 0.00352 0.00362 0.00393 0.00349 

0.25% 0.00293 0.00145 0.00121 0.00133 0.00130 0.00167 0.00137 

Dow Jones 
1.00% 0.00220 0.00078 0.00056 0.00073 0.00065 0.00080 0.00067 

0.25% 0.00044 0.00012 0.00003 0.00004 0.00003 0.00008 0.00006 

CAC40 
1.00% 0.00568 0.00602 0.00462 0.00445 0.00427 0.00445 0.00443 

0.25% 0.00282 0.00375 0.00185 0.00158 0.00148 0.00175 0.00178 

IBEX35 
1.00% 0.00554 0.00742 0.00308 0.00355 0.00336 0.00366 0.00350 

0.25% 0.00274 0.00516 0.00152 0.00161 0.00158 0.00186 0.00182 

Hang Seng 
1.00% 0.00333 0.00581 0.00048 0.00124 0.00127 0.00165 0.00125 

0.25% 0.00062 0.00128 0.00000 0.00001 0.00000 0.00006 0.00001 

Tel Aviv 
1.00% 0.00150 0.00270 0.00024 0.00060 0.00054 0.00069 0.00062 

0.25% 0.00030 0.00153 0.00000 0.00000 0.00000 0.00004 0.00003 

Ftsie100 
1.00% 0.00376 0.00399 0.00254 0.00227 0.00205 0.00228 0.00228 

0.25% 0.00126 0.00131 0.00056 0.00036 0.00029 0.00047 0.00048 

Note: This table reports the average of the loss function of each VaR model in both confidence levels. The average was 
multiplied by 1,000. Boldface figures denote the minimum value for the average of the loss function for each index.  

 
 

Table 8. Magnitude of the firm’s loss function  

 
 level NORMAL MME ST SGT IHS SSD SGED 

Nikkei 1.00% 0.00054 0.00056 0.00062 0.00059 0.00059 0.00058 0.00059 

0.25% 0.00066 0.00068 0.00080 0.00076 0.00077 0.00074 0.00075 

Merval 
1.00% 0.00056 0.00052 0.00079 0.00065 0.00066 0.00062 0.00066 

0.25% 0.00068 0.00063 0.00112 0.00090 0.00092 0.00081 0.00085 

S&P 500 
1.00% 0.00044 0.00046 0.00052 0.00051 0.00050 0.00049 0.00051 

0.25% 0.00054 0.00056 0.00066 0.00065 0.00065 0.00062 0.00064 

Dow Jones 
1.00% 0.00040 0.00044 0.00048 0.00046 0.00047 0.00045 0.00046 

0.25% 0.00050 0.00054 0.00062 0.00060 0.00061 0.00058 0.00059 

CAC40 
1.00% 0.00111 0.00120 0.00121 0.00122 0.00123 0.00122 0.00122 

0.25% 0.00136 0.00144 0.00150 0.00153 0.00154 0.00150 0.00150 

IBEX35 
1.00% 0.00109 0.00118 0.00132 0.00125 0.00127 0.00124 0.00125 

0.25% 0.00132 0.00144 0.00173 0.00167 0.00168 0.00158 0.00159 

Hang Seng 
1.00% 0.00062 0.00067 0.00080 0.00072 0.00071 0.00069 0.00071 

0.25% 0.00077 0.00081 0.00107 0.00092 0.00096 0.00089 0.00092 

Tel Aviv 
1.00% 0.00040 0.00041 0.00052 0.00046 0.00047 0.00045 0.00046 

0.25% 0.00050 0.00051 0.00069 0.00062 0.00062 0.00058 0.00059 

Ftsie100 
1.00% 0.00099 0.00110 0.00108 0.00111 0.00113 0.00110 0.00110 

0.25% 0.00122 0.00133 0.00135 0.00140 0.00143 0.00137 0.00136 
Note: This table reports the average of the loss function of each VaR model in both confidence levels. Boldface 
figures denote the minimum value for the average of the loss function for each index. 



 

 

20 

 Figure 1. Stock index returns 

NIKKEI

-15

-10

-5

0

5

10

15

2000 2002 2004 2006 2008 2010 2012

 

MERVAL

-20

-10

0

10

20

2000 2002 2004 2006 2008 2010 2012

 

S&P 500

-15

-10

-5

0

5

10

15

2000 2002 2004 2006 2008 2010 2012

 

DOW JONES

-10

-5

0

5

10

15

2000 2002 2004 2006 2008 2010 2012

 

CAC-40

-15

-10

-5

0

5

10

15

2000 2002 2004 2006 2008 2010 2012

 

IBEX-35

-15

-10

-5

0

5

10

15

2000 2002 2004 2006 2008 2010 2012

 

HANG  SENG

-20

-10

0

10

20

2000 2002 2004 2006 2008 2010 2012

 

TEL AVIV

-10

-5

0

5

10

15

2000 2002 2004 2006 2008 2010 2012

 

FTSIE-100

-10

-5

0

5

10

2000 2002 2004 2006 2008 2010 2012

 
This figure illustrates the daily evolution of returns of nine indexes (Nikkei, Merval, S&P 500, Dow Jones Industrial Average, CAC40, 
IBEX35, Hang Seng, Telaviv and Ftsie-100.) from January 3rd 2000 to November 30th, 2012. Source: Bloomberg. 

Figure 2. Volatility of the returns 

NIKKEI

0.000

0.005

0.010

0.015

0.020

2000 2002 2004 2006 2008 2010 2012

 

MERVAL

0

0.01

0.02

0.03

2000 2002 2004 2006 2008 2010 2012

 

S&P 500

0.000

0.004

0.008

0.012

2000 2002 2004 2006 2008 2010 2012

 

DOW JONES

0.000

0.004

0.008

0.012

2000 2002 2004 2006 2008 2010 2012

 

CAC-40

0.000

0.004

0.008

0.012

2000 2002 2004 2006 2008 2010 2012

 

IBEX-35

0.000

0.005

0.010

0.015

0.020

2000 2002 2004 2006 2008 2010 2012

 

HANG SENG

0.000

0.005

0.010

0.015

0.020

2000 2002 2004 2006 2008 2010 2012

 

TEL AVIV

0

0.002

0.004

0.006

0.008

0.01

2000 2002 2004 2006 2008 2010 2012

 

FTSIE-100

0.000

0.003

0.006

0.009

2000 2002 2004 2006 2008 2010 2012

 

Note: This figure illustrates the conditional volatility of daily returns. The volatility was estimated using the approach proposed by Franses  
and van Dijk (1999). Sample runs from January 3rd 2000 to November 30th, 2012. Source: Bloomberg. 



 

 

21 

 
Figure 3. Histograms, Normal versus other skewed distributions  

 

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
0

100

200

300

400

500

600

 

 

Nikkei
SGT
Normal

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
0

100

200

300

400

500

600

700

 

 

Nikkei
SGED
Normal

 

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
0

100

200

300

400

500

600

 

 

Nikkei
SSD
Normal

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
0

100

200

300

400

500

600

 

 

Nikkei
IHS
Normal

 

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
0

100

200

300

400

500

600

 

 

Nikkei
ST
Normal

 
Note: These figures illustrate the histograms, Normal distribution (blue line) versus the rest of considered 
distributions (red line). The data used in the graphs are those obtained from the Nikkei Index and the sample runs 
from January 3, 2000 to November 30, 2012.